Consistent invertibility and perturbations of property $R$

Let $B(X)$ be the space of all bounded linear operators on an infinite-dimensional complex Banach space $X$. An operator $T\in B(X)$ is said to be consistent invertibility if for arbitrary $S\in B(X)$, $TS$ and $ST$ are either both or neither invertible. Using induce spectrum, the paper gives the necessary and sufficient conditions for the stability of property $(R)$ under commuting power finite rank perturbations. Moreover, the paper studies the transmission of property $(R)$ from $T$ to $f(T)$ for any analytic function $f$ on a neighborhood of $\sigma(T)$. As an application, the paper proves that every generalized scalar operator satisfies property $(R)$ under commuting power finite rank perturbations.